Capacitance
Capacitors: Electrical device that store electric
charges
A capacitor consists of two conductors carrying
charges of equal magnitude but opposite sign separated
by an insulator or dielectric medium.
Capacitance: Depends on its geometry and on the
material that separates the conductors.
A potential difference dV exists between the
conductors due to the presence of the charges.
What is the capacity of the device for storing charge at
particular value of dV?
Capacitance
Experiments show the quantity of electric charge
Q on a capacitor is linearly proportional to
the potential difference between the
conductors, that is Q  V. Or we write Q = C
V
The capacitance C of a capacitor is the ratio of the
magnitude of the charge on either conductor to the
magnitude of the potential difference between them:
C 
Q
V
SI Unit: farad (F), 1F = 1 C/V
Typical device have capacitances ranging from microfarad to
picofarad.
Parallel - Plate Capacitors
A parallel-plate capacitor consists
of two parallel conducting
plates, each of area A, separated
by a distance d.
When the capacitor is charged,
the plates carry equal amounts of
charge. One plate carries positive
charge, and the other carries
negative charge.
The plates are charged by connection to a battery.
Let two parallel metallic plates of
equal area A separated by a distance
d as shown.
One plate carries a charge Q and the
other carries a charge –Q. And
surface charge density of each plate
is s = Q/A.
d
A
Variation with A
If plates are large, then charges can distribute themselves
over a substantial area, and the amount of charge that can
be stored on a plate for a given potential diff increases as
‘A’ is increased.
Thus we expect ‘C’ to be proportional to ‘A’.  C  A
Variation with d:
d
Potential difference V constant
across the plates, E field increases as
d decreases.
V  Ed
Imagine d decreases and consider
situation before any charges have had
a chance to move in response to this
change.
Because no charge move  E the
same over a shorter distance.
V = Ed means that V decreases.
A
The difference between this new capacitor
voltage and the terminal voltage of the
battery now exists as a potential difference
across the wires connecting the battery to
the capacitor.
A E field result in the wires that drives more
charge onto the plates, increasing the
potential diff. V until it matches that of the
battery.  potential diff. across wire = 0 
flow of charges stop.
d
A
More charges has accumulated at the capacitor as a result.
We have d decrease, Q increases. Similarly d increases 
Q decreases.
Capacitance inversely proportional to d.  C  1/d
The electric field between the plates of a parallel-plate
capacitor is uniform near the center but nonuniform
near the edges.
Formula of capacitance for a
Parallel-Plate Capacitors
Assume electric field uniform between the plates,
(from Gauss’s Law):
E 


0
Q
0A
V  Ed 
C 
Q
V

C 
Q
0A
d
Q
Qd /  0 A
0A
d
What is a lightning discharge ?
Friction forces between the air molecules
within a cloud result in positively charged
molecules moving to the lower surface,
and the negative charges moving to the
upper surface .
The lower surface induces a high
concentration of negative charges in the
earth beneath.
The resultant electric field is very strong,
containing large amounts of charge and
energy.
If the electric field is greater than
breakdown strength of air, a lightning
discharge occurs, in which air molecules
are ripped apart, forming a conducting
path between the cloud and the earth.
Cylindrical Capacitors
L
A solid cylindrical conductor of radius a and charge Q
is coaxial with a cylindrical shell of negligible
thickness, radius b > a, and charge –Q.
If we assume an ideal cylindrical capacitor:
•The electric field points radially from the inner cylinder
to the outer cylinder.
•The electric field is zero outside the collinear cylinders.
Assume that L is >> a and b,
neglect the end effects.
E is perpendicular to the long
axis of the cylinders and is
confined to the region between
them.
L
Potential difference between the
two cylinders is given by
b
V b  V a    E .dl
Where E is the field in the region a < r < b.
a
Our discussion on Gauss’s Law  E = 2k/r = 2kQ/rL
where  is the linear charge density of the cylinder.
Note that the charge on outer cylinder does not contribute
to E field inside it.
b
V b  V a    2kQ / rL.dl
a

2kQ
L
C 
Q
V

L
b
 1 / r.d l
a
L
b
2kln  
a
What is the capacitance per unit length ?
Q. Co-axial Cable. Read the cable, typically 50 pF/m.
Is this sensible ? Typically a  0.5 mm, b  1.5 mm
C/L 
1
2  8 . 99  10  ln( 3 )
9
 50pF/m
The Spherical Capacitors
A spherical capacitor consists of a spherical conducting
shell of radius b and charge –Q concentric with a
smaller conducting sphere of radius a and charge Q.
Find the capacitance of this device.
Gauss’s law  E field outside a
spherically
symmetric
charge
distribution is radial and given by
Er = kQ/r2.
The potential difference between
the spheres is
b
b
V b  V a    E .dr
 kQ 
a
a
C 
Q
Vb  V a

 1 1 
 kQ 
 
a
 b
dr
r
2
ab
k (b  a )
Capacitance of an isolated sphere
Q. Calculate the capacitance of an isolated
spherical conductor of radius R and charge Q
by assuming that the second conductor making
up the capacitor is a concentric hollow sphere
of infinite radius.
Q
Ans. Electric potential of the sphere of radius R is
kQ/R and V= 0 at infinity, we have
C 
Q
V

Q
kQ / R

R
k
 4 0 R
C is proportional to its radius and independent of both
the charge on the sphere and the potential difference.
Capacitors in Circuits
• Capacitors can be wired together in circuits in
parallel or series
– Capacitors in circuits connected by wires such
that the positively charged plates are connected
together and the negatively charged plates are
connected together, are connected in parallel.
– Capacitors wired together such that the
positively charged plate of one capacitor is
connected to the negatively charged plate of the
next capacitor are connected in series.
Capacitors in Parallel
Consider an electrical circuit with three capacitors wired
in parallel
• Each of three capacitors has one plate connected to the
positive terminal of a battery with voltage V and one
plate connected to the negative terminal.
.. key point for capacitors in parallel
• potential difference V across each capacitor is the
same.
• We can write the charge on each capacitor as
q1  C1V ,
q 2  C 2V ,
q 3  C 3V
q  q1  q 2  q 3  C 1V  C 2 V  C 3V  C 1  C 2  C 3 V
We can consider the three capacitors as one equivalent
capacitor Ceq that holds a total charge q given by
q  C eq V
C eq  C 1  C 2  C 3
n
A general result for n capacitors in
parallel is:
C eq 
C
i
i 1
Conclusion: If we can identify capacitors in a circuit that
are wired in parallel, we can replace them with an
equivalent capacitance.
Capacitors in Series
key point for capacitors in series:
The battery produces an equal charge q on each
capacitor because the battery induces a positive charge
on the positive place of C1, which induces a negative
charge on the opposite plate of C1, which induces a
positive charge on C2, etc.
Knowing that the charge is the same on all three
capacitors we can write
 1
1
1 
V  V1  V 2  V 3 


 q


C1 C 2
C3
C 3 
 C1 C 2
q
q
q
V 
q
C eq
We can express an equivalent capacitance Ceq as
1
C eq

1
C1

1
C2

1
C3
• We can generalize to n capacitors in series
1
C eq
n

1
C
i 1
i
• If we can identify capacitors in a circuit that are
wired in series, we can replace them with an
equivalent capacitance.